Tiles: an online algorithm for community discovery in dynamic social networks
Abstract
Community discovery has emerged during the last decade as one of the most challenging problems in social network analysis. Many algorithms have been proposed to find communities on static networks, i.e. networks which do not change in time. However, social networks are dynamic realities (e.g. call graphs, online social networks): in such scenarios static community discovery fails to identify a partition of the graph that is semantically consistent with the temporal information expressed by the data. In this work we propose Tiles, an algorithm that extracts overlapping communities and tracks their evolution in time following an online iterative procedure. Our algorithm operates following a domino effect strategy, dynamically recomputing nodes community memberships whenever a new interaction takes place. We compare Tiles with stateoftheart community detection algorithms on both synthetic and real world networks having annotated community structure: our experiments show that the proposed approach is able to guarantee lower execution times and better correspondence with the ground truth communities than its competitors. Moreover, we illustrate the specifics of the proposed approach by discussing the properties of identified communities it is able to identify.
Keywords
Community discovery Dynamic networks Social network analysis1 Introduction
In dynamic networks, the rise of new nodes and edges produces deep topological mutations and creates new paths connecting once disconnected components. Therefore, an algorithm that considers social networks as static entities—frozen in time—necessarily introduces bias on its results. For these reasons, we advocate the need to weaken the QSSA and imagine social networks as complex, mutable, evolving objects which change in an fluid manner every time a new interaction appears (or disappears). In this scenario, pursuing an online community discovery approach enables valuable complementary benefits such as: (1) the reduction of computational complexity (both in space and time), (2) the tracking of community dynamics, (3) the possibility to feed predictive models with punctual and fine grained information regarding how the network topology changes over time.
Nonetheless, a dynamic approach to community discovery enables interesting practical applications. For instance, timeaware approaches can be used by mobile phone carriers that want to propose a flexible billing plan for their customers by lowering call prices to users in the same social circle: indeed, when imposing a fixed network structure such marketing strategy looses its effectiveness since static communities overestimate or underestimate the real connectivity (Fig. 1). In contrast, a dynamic community discovery algorithm provides uptodate communities and helps the company in providing its users with a more customized service.
In this work, we propose to adapt the classical community discovery problem to the dynamic scenario and introduce an evolutionary formulation able to deal with evolving networks. This dynamic perspective on the community discovery problem allows us to investigate, describe and quantify relevant processes that take place on social networks, such as the evolution through time of the network community structure, the evolution through time of each single community both in terms of topology and events (birth, growth, death etc.) and even the evolution of single individuals connections within different communities. Moreover, we propose Tiles,^{1} an algorithm that tracks the evolution of communities through time. Our approach proceeds in a streaming fashion considering each topological perturbation as a fall of a domino tile: every time a new interaction appears in the network, Tiles first updates the communities locally and then propagates the changes to the node surroundings adjusting the neighbors’ community memberships. The online nature of Tiles brings many advantages. First, the computation of network substructures is local and involves a limited number of nodes and communities, thus speeding up the updating process. Second, our approach allows to observe two types of evolutionary behaviors: (1) the stability of individuals’ affiliations to communities, and (2) the evolution over time of interactionbased communities.
We validate the effectiveness of our algorithm by comparing it with stateoftheart community discovery algorithms, using both synthetic and real networks enriched by ground truth communities. In our experimental analysis we underline that Tiles is able to achieve a better match with the ground truth communities than the compared algorithms. Moreover, we show that Tiles guarantees lower execution times than the competitors since it can be easily parallelized. We also provide a characterization of the communities extracted by our algorithm by analyzing three Big Data sources: a nationwide call graph of one million users whose interactions are tracked for one month; a Facebook interaction network which covers a period of 52 weeks; an interaction network of 8 million users of the Chinese microblogging platform WEIBO observed for 1 year.
The paper is organized as follows. Section 2 summarizes the related works in community discovery, dynamic network analysis and evolutionary community discovery; Section 3 formalizes the problem of Evolutionary Community Discovery. Section 4 describes Tiles providing algorithmic details and showing some characteristics of the algorithm. In Sect. 5 we compare Tiles with other community detection algorithms and present a characterization of discovered communities. Finally, Sect. 6 concludes the paper, describing some scenarios for future works.
2 Background and related works
The problem of finding and tracking communities in an evolutionary context is relatively novel. Here we discuss some relevant works regarding classical community discovery, dynamic networks analysis and, as their merging point, evolutionary community discovery.
2.1 Community discovery
The problem of finding communities in complex networks is a hot topic, as witnessed by the high number of works in this field. A survey by Fortunato (2010) explores all the most popular techniques to find communities in complex networks. The more recent survey by Coscia et al. (2011) tries to classify families of algorithms based on the typology of the extracted communities. The classic definition of community relates to a dense subgraph, in which the number of edges among its nodes is significantly higher than the number of outgoing edges. However, this definition does not cover many real world scenarios, and many different alternative definitions of communities have been proposed. One of the most famous is based on the modularity concept, a quality function of a graph partition proposed by Clauset et al. (2004), which scores high values for partitions whose internal cluster density is higher than the external density. An alternative approach is the application of information theoretic techniques, as for example in Infomap (Rosvall and Bergstrom 2008). An interesting property for community discovery is the ability to return overlapping substructures, i.e., to allow nodes to be part of more than one community. This property reflects the social intuition that each person is part of multiple different communities (e.g. work, family, hobby...). A wide set of algorithms were developed over this property, such as the one proposed in Palla et al. (2005). Other overlapping approaches are based on Label Propagation such as Demon Coscia et al. (2012), a framework which allows a bottom–up formation of communities exploiting egonetworks. Given the rising interests on multiplex (multidimensional/multirelational) networks, recently some community discovery algorithms able to partition labeled multigraph have been proposed (Boden et al. 2012).
2.2 Dynamic network analysis
Several graph problems are, by their nature, closely tied to network dynamics (Kostakos 2009). The flowing of time plays different roles over a complex network: it can determine the evolution of the graph topology (e.g. edges and node can fall and rise, communities born and die) or lead to the observation of diffusion processes. Among the problems related to network evolution, Link Prediction is one of the most studied: formulated by Nowell and Kleinberg (2003) its aim is to predict edges that will appear in the future given the actual state of the network. Models for network growth, as the ones proposed in Barabási and Albert (1999) and Leskovec et al. (2005), replicate network evolutions peculiarity in order to build synthetic graphs. Furthermore, diffusion processes have been studied in order to understand virus epidemics (Wang et al. 2009) and spreading of innovations (Burt 1987).
2.3 Evolutionary community discovery
Communities are certainly the mesoscale structures most affected by changes in network topology: as time goes by the rise and fall of nodes and edges determines the appearance and vanishing of social clusters that static community discovery algorithms are unable to detect. In order to understand how communities evolve, three main approaches have been followed so far: Independent Community Detection and Matching, Global Informed iterative community detection, and Local Informed iterative community detection. In the following, we report a survey on these categories of approaches.
2.3.1 Independent community detection and matching
Strategies that fall in this category are prevalently aimed to track the evolution of communities by identifying key actions which regulate their life (birth, death, merge, split). Nguyen proposes an extended lifecycle model able to track, in an offline fashion, the evolution of communities (Nguyen 2012). Such methodology, as well as the one introduced in Goldberg et al. (2011), Dhouioui and Akaichi (2014), Takaffoli et al. (2014) and Asur et al. (2009), works on a twostep procedure: (1) the graph is divided in n temporal snapshots and, for each of them, a set of communities is extracted; (2) for each community an evolutionary chain is built by observing its evolution through temporal adjacent sets. In their work, Takaffoli et al. introduced Modec (Takaffoli et al. 2011), a framework able to model and detect the evolution of communities obtained at different snapshots in a dynamic social network. The problem of detecting the transition of communities is solved by identifying events that characterize the changes of the communities across time. Unlike previous approaches (Palla et al. 2007) the Modec framework is independent from the static community mining algorithm chosen to partition timestamped networks.
2.3.2 Global informed iterative community detection
A different methodology to detect communities in a dynamic scenario, is to design a procedure where each community identified at time t is influenced by the ones detected at time \(t1\) avoiding the need to match communities, thus introducing global smoothness in the community identification process. The approaches belonging to this category derive from the evolutionary clustering analysis (Chakrabarti et al. 2006). Folino and Pizzuti (2014) propose an evolutionary multiobjective approach to community discovery in dynamic networks which, moving from an evolutionary clustering perspective, searches for smooth community transitions among consecutive time steps. Rozenshtein et al. (2014) focus on identifying the optimal set of time intervals to discover dynamic communities in interaction networks. Although those approaches reduce the complexity of the matching phase, they are based on a static temporal partition of the complete temporal network. Other works belonging to this category are Sun et al. (2010), Shang et al. (2012) and Guo et al. (2014).
2.3.3 Local informed iterative community detection
The last category, also known as online approaches, is defined by algorithms that do not partition the full temporal annotated graph, but try to build and maintain communities in an online fashion following the rising and vanishing of new nodes and edges. Only a few works, at the best of our knowledge, have exploited this strategy so far. Qi et al. (2013) propose a probabilistic approach to determine dynamic community structure in a social sensing context. The main objective of the introduced ICDRF model is to dynamically maintain a community partition of moving objects based on trajectory information up to the current timestamp. However, due to the information used to update the community membership, the approach is suitable only for a specific kind of networked data. Lin et al. (2008) propose an iterative algorithm that, avoiding the classical twostep analysis, extract communities taking care of the topology of the graph at the specific time frame t as well as the historical evolutive patterns of previously computed communities. In Cazabet et al. (2010), Cazabet introduces iLCD an overlapping online approach to community detection which reevaluates communities at each new interaction according to the path lengths between each node and its surrounding communities. Xu et al. (2013) propose an algorithm aimed at analyzing the evolution of community cores. The proposed approach tracks only stable links within face to face interaction graphs exploiting a rule based online approach. Other works belonging to this category are Zakreweska and Bader (2015), Lee et al. (2014) and Nguyen et al. (2011).
Tiles belongs to the latter family of approaches. However, unlike the previously mentioned algorithms, it uses only local topological information and a constrained label propagation in order to minimize the computation needed to maintain updated the community structure.
3 Evolutionary community discovery
The overwhelming number of papers proposed in recent years clearly expresses that researchers are not interested in formulating “The Community Discovery algorithm” but in finding the right algorithm for each specific declination of the problem. Moving from this observation we tackle a specific and not yet deeply studied problem: evolutionary community discovery in dynamic social networks.
Definition 1
(Evolutionary Community Discovery) Given an interaction streaming source S and a graph \(G=(V,E)\), where \(e\in E\) is a triple (u, v, t) with \(u,v\in V\) and \(t\in {\mathbb {N}}\) is the time of the interaction’s generation by S, the Evolutionary Community Discovery (ECD) aims to identify and maintain updated the community structure of G as new interactions are generated by S.
The source S produces new interactions among pair of nodes which can be either already part of the graph or newcomers. It models scenarios in which interactions do not occur with a rigid temporal discretization but flow “in streams” as time goes by. After all, this is how social interactions actually take place: phone calls, SMS messages, tweets, Facebook posts are produced in a fluid streaming fashion and consequently the corresponding networks’ social communities also change fluidly over time. In contrast with a static community detection algorithm, an ECD algorithm must produce a series of communities’ observations in order to describe how time shapes network topologies in coherent substructures. Moreover, an ECD algorithm should address the following question: given a community C at timestamp t and a streaming source S, what its structure will be at an arbitrary time \(t+\Delta \) given the interactions produced by S? In order to answer this question the discovery process must be able to smooth the evolution of a community from t to \(t+\Delta \) by identifying its local mutations avoiding external matching as done by traditional twostep approaches. Our algorithm, Tiles, is designed to solve the ECD problem since it tracks the evolution of communities following a domino tile strategy.
4 Tiles algorithm
Social interactions determine how communities form and evolve: indeed, the rising and vanishing of interactions can change the communities’ equilibrium. A common approach in literature to address topology dynamics is to: (1) split the network into temporal snapshots, (2) repeat a static community detection for each snapshot and (3) study the variation of the results as time goes by. This approach introduces an evident issue: which temporal threshold has to be chosen to partition the network? This problem, which is obviously context dependent, also introduces another one: once the algorithm is performed on each snapshot how can we identify the same community in consecutive time slots? To overcome these issues we propose Tiles, an ECD algorithm that does not impose fixed temporal thresholds for the partition of the network and the extraction of communities. It proceeds analyzing an interaction stream: every time a new interaction is produced by a given streaming source, Tiles uses a label propagation procedure to diffuse the changes to the node surroundings and adjust the neighbors’ community memberships. A node can belong to a community with two different levels of involvement: peripheral membership and core membership. If a node is involved in at least a triangle with other nodes in the same community it is a core node while if it is an onehop neighbor of a core node it is a peripheral node. Only core nodes are allowed during the label propagation phase to spread community membership to their neighbors. Tiles generates overlapping communities, i.e. each node can belong to different communities which can represent the different spheres of the social world of an individual (friendship, working relations, etc.).
The algorithm^{2} takes as input four parameters: (1) the graph G, which is initially empty; (2) an edge streaming source S; (3) \(\tau \), a temporal observation threshold; (iv) a Time To Leave (ttl) value for the interactions. The temporal observation threshold \(\tau \) specifies how often we want to observe the structure of the communities allowing us to customize the output of the algorithm. Furthermore, ttl models the expected lifespan of a new interaction: it acts as a temporal decreasing countdown that, when expired, leads to the removal of the edge it is attached to. Indeed the value of ttl impacts the overall stability of the observed phenomena.
 1.
both nodes u and v appear for the first time in the graph. No other actions are performed until the next interaction is produced by the source S (Fig. 2a, Algorithm 1 lines 10–12);
 2.
one node appears for the first time and the other is already existing but peripheral or both nodes are existing but peripheral, in any case they do not belong to any community core. Since peripheral nodes are not allowed to propagate the community membership, no action is performed (none of the “if” clauses is satisfied) until the next interaction is produced by the source S (Fig. 2a, Algorithm 1 lines 14–18);
 3.
one node appears for the first time in G while the other is an already existing core node. The new node inherits a peripheral community membership from the existing core node (Fig. 2b, Algorithm 1 lines 20–23);
 4.both nodes are core nodes already existing in G (Algorithm 1 lines 25–33). In this case we have two possible subscenarios:
 (a)
Nodes u and v do not have common neighbors (we identify as \(\Gamma (u)\) the set of neighbors of u): they propagate each other a peripheral community membership through the PeripheralPropagation procedure (Fig. 2c, Algorithm 1 lines 27–29);
 (b)
Nodes u and v do have common neighbors: their community memberships are reevaluated and the changes propagated to their surroundings by the CorePropagation function (Fig. 2c, d, Algorithm 1 lines 30–32).
 (a)
The PeripheralPropagation procedure regulates the events where a new node becomes part of an already established community. Since, initially, the newcomer is not involved in any triangle with other nodes of the community it becomes part of its periphery. The same function is performed when a new interaction connects existing nodes that do not share any neighbors.
4.1 Expired edges removal
 1.
(u, v, t) is expired: the edge is removed from the graph and from \({ RQ}\) (Algorithm 4, lines 217);
 2.
(u, v, t) is still valid: since the interactions in \({ RQ}\) are ordered the execution of removeExpiredEdges is terminated (Algorithm 4, line 19).
 (a)
the original community is not “broken”, i.e. it is still composed by a single component: we need only to reevaluate the “roles” of nodes u, v and their first level neighbors (Algorithm 4, lines 8–9);
 (b)
the original community is splitted into two or more separate entities: each component is then considered as a new community and the “roles” of nodes are recomputed (Algorithm 4, lines 10–16).
Even if in the proposed formulation the interaction removal is performed through a fixed size sliding window controlled by the ttl parameter, Tiles can be easily parametrized to allow custom removal strategies. For instance, we can substitute the interaction validity check (Algorithm 4 line 2) with a decay function^{3} or to handle directly interaction removal as done for the insertion, i.e. leveraging if available in the data explicit information on edge vanishing. Certainly this latter scenario represents the optimum since the analyst does not need to define arbitrary thresholds and/or make additional assumptions. We choose to adopt a sliding window in order to provide a simple and tunable way to simulate asynchronous updates, an approach often used when dealing with temporal annotated data streams.
4.2 Computational complexity of Tiles
Since Tiles operates on streaming data its complexity analysis depends on two main operations: interaction insertion and removal.
4.2.1 Interaction insertion phase

both u and v appear for the first time: no action taken, so complexity O(1);

at least one node was already present in the network but both nodes are not core: no action taken, so complexity O(1);

node u is core in one or more communities and node v is new: PeripheralPropagation is called on the new node: the function cycles on the communities for which u is core to perform the propagation on v, so complexity \(O(core(v)) < O(V)\);

u and v are both core nodes for one or more communities that do not share neighbors: PeripheralPropagation is called on both nodes, so complexity \(O(core(v)+core(u)) < O(V)\);

u and v are core nodes for one or more communities that share neighbors: CorePropagation is called. This function cycles over the common neighbors of u and v, which in the worst case scenario are all the nodes of the network (O(V)), updates the community cores and performs PeripheralPropagation \(O(core(v)+core(u))\). Thus the final complexity is \(O(V*(core(u)+core(v)))\).
4.2.2 Interaction removal phase
 1.Main loop on the removal queue RQ (Algorithm 4). The cycle is executed until a valid interaction is found:

if \(ttl=0\) (zeromemory scenario) it consumes all the edges in RQ: therefore, we have O(E) cycles;

if \(0<ttl<\infty \), \(RQ_{ttl}\) is the expected average size of the interactions processed when removeExpiredEdges is called: we have \(O(RQ_{ttl}) < O(E)\) cycles;

if \(ttl=\infty \) (full memory scenario), the removal is not executed at all, thus the complexity is O(1).

 2.
Node role update (Algorithm 5). The main computational cost here is due to the clustering coefficient computation for each of the selected nodes on the communityinduced graph. A naive implementation, assuming a complete clique, has cubic cost on the cardinality of nodes to be updated \(O(to\_update^{3})\). Interaction networks are sparse so, in order to provide a more realistic complexity, we can assume that that every node in the set has \(\sqrt{to\_update}\) neighbors within the community: thus, we can estimate the overall complexity with \(O(\sqrt{to\_update}^2)*to\_update = O(to\_update^2)\).
4.3 Tiles properties
Given its streaming nature Tiles shows two main properties: (1) it can be used incrementally on a precomputed community set; (2) it can be parallelized if specific conditions are satisfied. Moreover, in presence of a deterministic interaction source S (i.e., a generator that always produces the same ordered sequence of interactions), the output of Tiles is uniquely determined. In this section we discuss and formalize such characteristics.
4.3.1 Incrementality
Incrementality is a property that an online algorithm operating on streaming data must satisfy: it assures that every new network perturbation produces updates in the community status and that the computation proceeds smoothly one interaction after the other. Moreover it ensures that the approach does not require external community matching across time as done by twostep approaches. Incrementality, if the stream source is deterministic, imposes that the final partition is univocally determined. In Tiles this is ensured by construction by the mutual exclusivity of the update rules: given a new edge and a given network status only a single pattern among the ones described can be executed (both for insertion and deletion). On the other hand in case of a nondeterministic streaming source, incrementality guarantees the smoothness of the updates but not that the final partition will be always the same (i.e., given a set of interactions generated with a different ordering w.r.t. a previous Tiles execution it is not assured to reach the same final partition).
4.3.2 Compositionality
5 Experimental results
Evaluating the results provided by a community detection algorithm is a hard task, since there is not a shared and universally accepted definition of what a community is. In literature each approach provides its own community definition, often maximizing a specific quality function (e.g. modularity, density, conductance, ...). Even though the communities identified by a given algorithm on a network are consistent with its community definition, it is not guaranteed that they are able to capture the real subtopology of the network. For this reason, a common methodology used to assess the quality of a community detection algorithm is to evaluate the similarity between the partition it produces with the ground truth communities of the analyzed network.
In this section, we compare Tiles to other stateoftheart algorithms on both synthetic and real networks with ground truth communities (Sect. 5.1). Moreover, we characterize the communities our algorithm produces on three largescale realworld datasets of social interactions (Sect. 5.2), discuss the eventbased community lifecycle of communities (Sect. 5.3) and, finally, analyze the impact of the ttl parameter on the node/community stability.
5.1 Evaluation on networks with ground truth communities
We compare Tiles with other static (Demon and cFinder) and dynamic (iLCD) overlapping community discovery algorithms. In order to cope with the absence of edge removal in the other algorithms we instantiate Tiles for an accumulative growth scenario (\(ttl = \infty \)). Demon (Coscia et al. 2012) is a bottomup approach which exploits labelpropagation to identify communities from egonetworks.^{4} cFinder (Palla et al. 2005) is an algorithm based on clique percolation that searches for cliquebased network structures.^{5} iLCD (Cazabet et al. 2010) is an algorithm for dynamic networks which reevaluates communities at each new interaction produced by a streaming source.^{6} In particular, every time a new interaction appears iLCD recomputes communities according to the path lengths between each node and its surrounding communities.
The slightly different community definitions introduced by the chosen algorithms make questionable a direct comparison of the outputs obtained on the same network when a ground truth is not provided. To overcome such issue and perform the analysis in a controlled environment we use both synthetic and real networks with ground truth communities.
5.1.1 Synthetic networks

N, the network size (from 1k to 500k nodes);

C, the network density (from 0 to 0.9, steps of 0.1);

\(\mu \), the average pernode ratio between the number of edges to its communities and the number of edges with the rest of the network (from 0 to 0.9, steps of 0.1).
5.1.2 Real networks
 Community Precision: the percentage of nodes in algorithm community x labeled with ground truth community y, computed as$$\begin{aligned} P=\frac{x\cap y}{x} \end{aligned}$$(4)
 Community Recall: the percentage of nodes in ground truth community y covered by algorithm community x, computed as$$\begin{aligned} R=\frac{x\cap y}{y}. \end{aligned}$$(5)
Real world network datasets
Network  Nodes  Edges  Coms.  CC  d  Tiles  iLCD  cFinder  Demon 

Amazon  334,863  925,872  75,149  .396  44  .78(.05)  .78(.23)  .77(.27)  .75(.24) 
Dblp  317,080  1,049,866  13,477  .632  21  .80(.09)  .70(.23)  .74(.24)  .65(.24) 
Youtube  1,134,890  2,987,624  8385  .080  20  .64(.11)  .42(.20)  .60(.20)  .42(.10) 
LiveJournal  3,997,962  34,681,189  287,512  .284  17  .73(.17)  .71(.04)  .32(.30)  .64(.29) 
5.2 Characterization on largescale real networks
Once compared Tiles with stateoftheart approaches both in synthetic and real world data, in this subsection we analyze the communities it produces on three largescale realworld dynamic interaction networks: a wall post network extracted from Facebook, a Chinese microblogging mention network, and a nationwide call graph extracted from mobile phone data.
General features of the networks
Network  Nodes  Edges  CC  #Observations (\(\tau \)) 

CG  1,007,567  16,276,618  0.067  10 (3 days) 
FB07  19,561  304,392  0.104  52 (1 week) 
 8,335,605  49,595,797  0.014  52 (1 week) 
5.2.1 Call graph
The call graph is extracted from a nationwide mobile phone dataset collected by a European carrier for billing and operational purposes. It contains date, time and coordinates of the phone tower routing the communication for each call and text message sent by 1, 007, 567 anonymized users during one month. We discarded all the calls to external operators. In the experiments we adopt as \(\tau \) a window of 3 days.
5.2.2 Facebook wallpost
The FB07 network is extracted from the WOSN2009 (Viswanath et al. 2009) dataset^{10} and regards online interactions between users via the wall feature in the New Orleans regional network during 2007. We adopted an observation period \(\tau \) of 1 week.
5.2.3 WEIBO interactions
This dataset is obtained from the 2012 WISE Challenge^{11}: built upon the logs of the popular Chinese microblog service WEIBO,^{12} its interactions represent mentions of users in short messages. We selected a single year, 2012, and used an observation window of one week.
It is worth noting that any arbitrary chosen value of \(\tau \) does not affect the execution of Tiles but only the number and frequency of community status observation. The \(\tau \) threshold is introduced with the purpose of simplifying the analysis of results reducing the number of community observations. Note that in order to get as output the complete history of community updates (an observation for each local perturbation) it is sufficient to set \(\tau \) equal to the clock of the streaming source.
A peculiarity of Tiles is the concept of community periphery. As discussed in Section 4 peripheral nodes are not involved in triangles with other nodes of the community. Every node first joins a community as peripheral node then it becomes a core one once it is involved within a triangle with other core nodes. We found that the expected time of transition from the periphery to the core of a community is generally short (Fig. 10d, e, f, Transitions In): in CG 40 % of nodes become core nodes in just 3 days; in FB07 15 % of nodes perform the transition during the first week; in WEIBO almost 60 % of transitions occur within a single week. Moreover, the transitions of nodes from the core to the periphery (Fig. 10d, e, f, Transitions Out) follow distributions similar to the ones observed for the reverse path. However, if we do not consider the distributions shapes but the total number of both events an interesting pattern emerges: the number of nodes that are “attracted” by the core of communities are between 2 and 900 times more of than the nodes that follow the opposite route (263,483 to 8415 in FB07, 680,932 to 420,530 in CG and 6,373,316 to 7070 in WEIBO). This peculiarity highlights that the community cores are able to provide meaningful—and stable—boundaries around nodes that frequently interact with each other.
We also investigated how many nodes perform the transition from periphery to core across consecutive community observations: we asked ourself, given two observation of a community C, what is the ratio of core nodes in C at \(t+\Delta \) that where in the peripheral nodes at time t? In Fig. 10d, e, f dashed line, we observe that this ratio has values between the 30 % and 50 % of the nodes for CG and FB07, and around 70 % for WEIBO. This means that in all the networks almost the half of the peripheral nodes become core nodes in the subsequent time window.
5.3 Eventbased community lifecycle and time to leave analysis

Birth (B): the community first appearance, i.e. the rising of the first set of core nodes of the community;

Merge/Absorption: two or more communities merge when their core nodes completely overlap: we define as Absorbed (A) the communities which collide with an existing one, we define as Merged (M) the already existing community;

Split (S): a community splits in one or more subcommunities as consequence of the edge removal phase;

Death (D): a community dies when its core node set becomes empty.

higher ttl values produce bigger communities and foster the stabilization of the node memberships;

lower ttl values produce smaller, denser, and often more unstable communities.
6 Conclusion and future works
In this paper we proposed Tiles, a community discovery algorithm which tracks the evolution of overlapping communities in dynamic social networks. It follows a “domino” approach: each new interaction determines the reevaluation of community memberships for the endpoints and their neighborhoods. Tiles defines two types of community memberships: peripheral membership and core membership, the latter indicating nodes involved in at least a triangle within the community. An interesting property of Tiles is compositionality, which allows for algorithm parallelization, thus speeding up the computation of the communities. Other interesting characteristics emerged by the application of the algorithm on largescale realworld dynamic networks, such as the skewed distribution of Tiles community size and their high average clustering coefficient. Compared with other community detection algorithms both on synthetic and real networks, Tiles shows better execution times and a higher correspondence with the ground truth communities. Moreover we shown how our approach enables the identification of the main events regulating the community lifecycle (i.e., birth, merge, split and death).
Many lines of research remains open for future works, such as identifying a more complex and precise way to manage the removal phase: indeed, one limit of the current approach is the needs of defining explicitly a time to leave threshold that is the same for all the interactions among the nodes within the social network. To overcome this issue we plan to define a data driven approach able to dynamically provide an estimate of the expected persistence for each single interaction. Moreover, the mechanisms which regulate the node transitions from the periphery to the core of a community is another interesting aspect we propose to investigate: once fully understood it can be exploited as predictive information in a link prediction scenario, or used to explore how the transition of nodes from periphery to core can affect the spreading of information over the network.
Footnotes
 1.
Temporal Interactions a Local Edge Strategy.
 2.
Tiles Python implementation available at: https://github.com/GiulioRossetti/TILES
 3.
 4.
Demon Python implementation available at: https://github.com/GiulioRossetti/DEMON.
 5.
cFinder C implementation available at: http://www.cfinder.org/.
 6.
iLCD Java implementation available at: http://cazabetremy.fr/iLCD.html.
 7.
All the algorithms were executed on a Linux 3.12.0 machine with an Intel Core i72600 CPU @3.4GHzx8 at 3.2GHz and 8GB of RAM.
 8.
NF1 Python code available at: https://github.com/GiulioRossetti/f1communities
 9.
The networks are available at: https://snap.stanford.edu/data/.
 10.
 11.
 12.
 13.
FB07 and CG show similar behaviors.
Notes
Acknowledgments
This work is partially Funded by the European Community’s H2020 Program under the funding scheme “FETPROACT12014: Global Systems Science (GSS)”, Grant agreement #641191 CIMPLEX “Bringing CItizens, Models and Data together in Participatory, Interactive SociaL EXploratories”, https://www.cimplexproject.eu. This work is supported by the European Community’s H2020 Program under the scheme “INFRAIA120142015: Research Infrastructures”, Grant agreement #654024 “SoBigData: Social Mining & Big Data Ecosystem”, http://www.sobigdata.eu.
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