Many dynamic networks coming from real-world contexts are link streams, i.e. a finite collection of triplets @math where @math and @math are two nodes having a link between them at time @math . A very large number of studies on these objects start by aggregating the data in disjoint time windows of length @math in order to obtain a series of graphs on which are made all subsequent analyses. Here we are concerned with the impact of the chosen @math on the obtained graph series. We address the fundamental question of knowing whether a series of graphs formed using a given @math faithfully describes the original link stream. We answer the question by showing that such dynamic networks exhibit a threshold for @math , which we call the , beyond which the properties of propagation of the link stream are altered, while they are mostly preserved before. We design an automatic method to determine the saturation scale of any link stream, which we apply and validate on several real-world datasets.
It is paradoxical to note that, while the question of the influence of aggregation on the properties of the formed graph series is largely ignored in most of the studies on dynamic networks, this question actually already received a lot of specific attention @cite_34 @cite_21 @cite_44 @cite_2 @cite_22 @cite_37 @cite_38 @cite_17 .
@cite_38 @cite_37 @cite_22 @cite_21 @cite_44 @cite_2 @cite_34 @cite_17
2293971114 2001667451 1723561712 2124637492
Given a network represented by a graph @math G=V,E, we consider a dynamical process of influence diffusion in G that evolves as follows: Initially only the nodes of a given @math S⊆V are influenced; subsequently, at each round, the set of influenced nodes is augmented by all the nodes in the network that have a sufficiently large number of already influenced neighbors. The question is to determine a small subset of nodes Sa target set that can influence the whole network. This is a widely studied problem that abstracts many phenomena in the social, economic, biological, and physical sciences. It is known [6] that the above optimization problem is hard to approximate within a factor of @math 2log1-∈|V|, for any @math ∈>0. In this paper, we present a fast and surprisingly simple algorithm that exhibits the following features: 1 when applied to trees, cycles, or complete graphs, it always produces an optimal solution i.e., a minimum size target set; 2 when applied to arbitrary networks, it always produces a solution of cardinality matching the upper bound given in [1], and proved therein by means of the probabilistic method; 3 when applied to real-life networks, it always produces solutions that substantially outperform the ones obtained by previously published algorithms for which no proof of optimality or performance guarantee is known in any class of graphs. Determining interrelatedness structure of various entities from multiple time series data is of significant interest to many areas. Knowledge of such a structure can aid in identifying cause and effect relationships, clustering of similar entities, identification of representative elements and model reduction. The majority of existing results are based on correlation based indices which effectively assume a static relationship between the time series data and are not suitable for detecting interrelatedness when the time series are dynamically related or when the time series involve loops. In this paper, a methodology for identifying the interrelatedness structure of dynamically related time series data is presented that also allows for the presence of loops in the connectivity structure. A linear dynamic graph model is presented where it is assumed that each time series data is the sum of an independent stochastic noise source and a dynamically weighted sum of other time series data. A link is assumed to be present between two time series if the weight of a time series, which is a linear time-invariant filter, is nonzero in the formation of the other. Reconstruction of the link connectivity structure under various scenarios is considered. It is shown that when the linear dynamic graph is allowed to admit non-causal weights, then the links structure can be recovered with the possibility of identifying spurious connections. However, it is shown that the spurious links remain local, where, a spurious link is restricted to be within one hop of a true link. Furthermore, strategies for exact reconstruction of the link structure when the weights are restricted to be causal are developed. The main tools for determining the network topology are based on variations of Wiener filtering. A significant insight provided by the article is that, in the class of network models considered in the paper, the Wiener filter estimating a stochastic process based on other processes remains local in the sense that the Wiener filter utilizes only measurements local to the node being estimated. Complex networks are often constructed by aggregating empirical data over time, such that a link represents the existence of interactions between the endpoint nodes and the link weight represents the intensity of such interactions within the aggregation time window. The resulting networks are then often considered static. More often than not, the aggregation time window is dictated by the availability of data, and the effects of its length on the resulting networks are rarely considered. Here, we address this question by studying the structural features of networks emerging from aggregating empirical data over different time intervals, focussing on networks derived from time-stamped, anonymized mobile telephone call records. Our results show that short aggregation intervals yield networks where strong links associated with dense clusters dominate; the seeds of such clusters or communities become already visible for intervals of around one week. The degree and weight distributions are seen to become stationary around a few days and a few weeks, respectively. An aggregation interval of around 30 days results in the stablest similar networks when consecutive windows are compared. For longer intervals, the effects of weak or random links become increasingly stronger, and the average degree of the network keeps growing even for intervals up to 180 days. The placement of the time window is also seen to affect the outcome: for short windows, different behavioural patterns play a role during weekends and weekdays, and for longer windows it is seen that networks aggregated during holiday periods are significantly different. The emergence of order in natural systems is a constant source of inspiration for both physical and biological sciences. While the spatial order characterizing for example the crystals has been the basis of many advances in contemporary physics, most complex systems in nature do not offer such high degree of order. Many of these systems form complex networks whose nodes are the elements of the system and edges represent the interactions between them. Traditionally complex networks have been described by the random graph theory founded in 1959 by Paul Erdohs and Alfred Renyi. One of the defining features of random graphs is that they are statistically homogeneous, and their degree distribution (characterizing the spread in the number of edges starting from a node) is a Poisson distribution. In contrast, recent empirical studies, including the work of our group, indicate that the topology of real networks is much richer than that of random graphs. In particular, the degree distribution of real networks is a power-law, indicating a heterogeneous topology in which the majority of the nodes have a small degree, but there is a significant fraction of highly connected nodes that play an important role in the connectivity of the network. The scale-free topology of real networks has very important consequences on their functioning. For example, we have discovered that scale-free networks are extremely resilient to the random disruption of their nodes. On the other hand, the selective removal of the nodes with highest degree induces a rapid breakdown of the network to isolated subparts that cannot communicate with each other. The non-trivial scaling of the degree distribution of real networks is also an indication of their assembly and evolution. Indeed, our modeling studies have shown us that there are general principles governing the evolution of networks. Most networks start from a small seed and grow by the addition of new nodes which attach to the nodes already in the system. This process obeys preferential attachment: the new nodes are more likely to connect to nodes with already high degree. We have proposed a simple model based on these two principles wich was able to reproduce the power-law degree distribution of real networks. Perhaps even more importantly, this model paved the way to a new paradigm of network modeling, trying to capture the evolution of networks, not just their static topology.