Silvio Lattanzi

On compressing social networks

Motivated by structural properties of the Web graph that support efficient data structures for in memory adjacency queries, we study the extent to which a large network can be compressed. Boldi and Vigna (WWW 2004), showed that Web graphs can be compressed down to three bits of storage per edge; we study the compressibility of social networks where again adjacency queries are a fundamental primitive. To this end, we propose simple combinatorial formulations that encapsulate efficient compressibility of graphs. We show that some of the problems are NP-hard yet admit effective heuristics, some of which can exploit properties of social networks such as link reciprocity. Our extensive experiments show that social networks and the Web graph exhibit vastly different compressibility characteristics.

Filtering: a method for solving graph problems in MapReduce

The MapReduce framework is currently the de facto standard used throughout both industry and academia for petabyte scale data analysis. As the input to a typical MapReduce computation is large, one of the key requirements of the framework is that the input cannot be stored on a single machine and must be processed in parallel. In this paper we describe a general algorithmic design technique in the MapReduce framework called filtering. The main idea behind filtering is to reduce the size of the input in a distributed fashion so that the resulting, much smaller, problem instance can be solved on a single machine. Using this approach we give new algorithms in the MapReduce framework for a variety of fundamental graph problems for sufficiently dense graphs. Specifically, we present algorithms for minimum spanning trees, maximal matchings, approximate weighted matchings, approximate vertex and edge covers and minimum cuts. In all of these cases, we parameterize our algorithms by the amount of memory available on the machines allowing us to show tradeoffs between the memory available and the number of MapReduce rounds. For each setting we will show that even if the machines are only given substantially sublinear memory, our algorithms run in a constant number of MapReduce rounds. To demonstrate the practical viability of our algorithms we implement the maximal matching algorithm that lies at the core of our analysis and show that it achieves a significant speedup over the sequential version.

Rumor spreading in social networks

Social networks are an interesting class of graphs likely to become of increasing importance in the future, not only theoretically, but also for its probable applications to ad hoc and mobile networking. Rumor spreading is one of the basic mechanisms for information dissemination in networks; its relevance stemming from its simplicity of implementation and effectiveness. In this paper, we study the performance of rumor spreading in the classic preferential attachment model of Bollobas et al. which is considered to be a valuable model for social networks. We prove that, in these networks: (a) The standard PUSH-PULL strategy delivers the message to all nodes within O(log^2n) rounds with high probability; (b) by themselves, PUSH and PULL require polynomially many rounds. (These results are under the assumption that m, the number of new links added with each new node is at least 2. If m=1 the graph is disconnected with high probability, so no rumor spreading strategy can work.) Our analysis is based on a careful study of some new properties of preferential attachment graphs which could be of independent interest.

SoK: The Evolution of Sybil Defense via Social Networks

Sybil attacks in which an adversary forges a potentially unbounded number of identities are a danger to distributed systems and online social networks. The goal of sybil defense is to accurately identify sybil identities. This paper surveys the evolution of sybil defense protocols that leverage the structural properties of the social graph underlying a distributed system to identify sybil identities. We make two main contributions. First, we clarify the deep connection between sybil defense and the theory of random walks. This leads us to identify a community detection algorithm that, for the first time, offers provable guarantees in the context of sybil defense. Second, we advocate a new goal for sybil defense that addresses the more limited, but practically useful, goal of securely white-listing a local region of the graph.

Almost tight bounds for rumour spreading with conductance

We show that if a connected graph with

Affiliation networks

n

Rumor Spreading in Social Networks

nodes has conductance φ then rumour spreading, also known as randomized broadcast, successfully broadcasts a message within ~O(φ^-1 • log n), many rounds with high probability, regardless of the source, by using the PUSH-PULL strategy. The ~O(••) notation hides a polylog φ^-1 factor. This result is almost tight since there exists graph of n nodes, and conductance φ, with diameter Ω(φ^-1 • log n). If, in addition, the network satisfies some kind of uniformity condition on the degrees, our analysis implies that both both PUSH and PULL, by themselves, successfully broadcast the message to every node in the same number of rounds.

Arrival and departure dynamics in social networks

In the last decade, structural properties of several naturally arising networks (the Internet, social networks, the web graph, etc.) have been studied intensively with a view to understanding their evolution. In recent empirical work, Leskovec, Kleinberg, and Faloutsos identify two new and surprising properties of the evolution of many real-world networks: densification (the ratio of edges to vertices grows over time), and shrinking diameter (the diameter reduces over time to a constant). These properties run counter to conventional wisdom, and are certainly inconsistent with graph models prior to their work. In this paper, we present the first model that provides a simple, realistic, and mathematically tractable generative model that intrinsically explains all the well-known properties of the social networks, as well as densification and shrinking diameter. Our model is based on ideas studied empirically in the social sciences, primarily on the groundbreaking work of Breiger (1973) on bipartite models of social networks that capture the affiliation of agents to societies. We also present algorithms that harness the structural consequences of our model. Specifically, we show how to overcome the bottleneck of densification in computing shortest paths between vertices by producing sparse subgraphs that preserve or approximate shortest distances to all or a distinguished subset of vertices. This is a rare example of an algorithmic benefit derived from a realistic graph model. Finally, our work also presents a modular approach to connecting random graph paradigms (preferential attachment, edge-copying, etc.) to structural consequences (heavy-tailed degree distributions, shrinking diameter, etc.).

Linking Users Across Domains with Location Data: Theory and Validation

Social networks are an interesting class of graphs likely to become of increasing importance in the future, not only theoretically, but also for its probable applications to ad hoc and mobile networking. Rumor spreading is one of the basic mechanisms for information dissemination in networks, its relevance stemming from its simplicity of implementation and effectiveness. In this paper, we study the performance of rumor spreading in the classic preferential attachment model of Bollobas et al. which is considered to be a valuable model for social networks. We prove that, in these networks: (a) The standard PUSH-PULL strategy delivers the message to all nodes within O (log2 n ) rounds with high probability; (b) by themselves, PUSH and PULL require polynomially many rounds. (These results are under the assumption that m , the number of new links added with each new node is at least 2. If m = 1 the graph is disconnected with high probability, so no rumor spreading strategy can work.) Our analysis is based on a careful study of some new properties of preferential attachment graphs which could be of independent interest.

Efficient Densest Subgraph Computation in Evolving Graphs

In this paper, we consider the natural arrival and departure of users in a social network, and ask whether the dynamics of arrival, which have been studied in some depth, also explain the dynamics of departure, which are not as well studied. Through study of the DBLP co-authorship network and a large online social network, we show that the dynamics of departure behave differently from the dynamics of formation. In particular, the probability of departure of a user with few friends may be understood most accurately as a function of the raw number of friends who are active. For users with more friends, however, the probability of departure is best predicted by the overall fraction of the users neighborhood that is active, independent of size. We then study global properties of the sub-graphs induced by active and inactive users, and show that active users tend to belong to a core that is densifying and is significantly denser than the inactive users. Further, the inactive set of users exhibit a higher density and lower conductance than the degree distribution alone can explain. These two aspects suggest that nodes at the fringe are more likely to depart and subsequent departure are correlated among neighboring nodes in tightly-knit communities.